Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach

Kathrin Hellmuth and Christian Klingenberg
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Kathrin Hellmuth: Department of Mathematics, University of Würzburg, 97074 Würzburg, Germany
Christian Klingenberg: Department of Mathematics, University of Würzburg, 97074 Würzburg, Germany

Mathematics, 2022, vol. 10, issue 3, 1-20

Abstract: In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.

Keywords: numerical finance; Black Scholes equation; uncertainty quantification; uncertain volatility; polynomial chaos; Bi-Fidelity method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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